VoC Analyses Part 3: Analyze Reinforcement-Learning Results

AIC analyses

Average AIC by age group

Average AIC

AIC difference from best model

Age-related change in parameter estimates from models

Run regressions between model parameters and age

## 
## Call:
## lm(formula = LL ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -141.012  -41.027   -2.953   37.260  141.220 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -265.726     25.786 -10.305  < 2e-16 ***
## age            3.982      1.386   2.873  0.00508 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 62.96 on 90 degrees of freedom
## Multiple R-squared:  0.08399,    Adjusted R-squared:  0.07382 
## F-statistic: 8.253 on 1 and 90 DF,  p-value: 0.005075

## 
## Call:
## lm(formula = alphaPosChoice ~ age, data = model_params)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.2864 -0.1951 -0.1085  0.1003  0.7838 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.123105   0.116696   1.055    0.294
## age         0.006673   0.006273   1.064    0.290
## 
## Residual standard error: 0.2849 on 90 degrees of freedom
## Multiple R-squared:  0.01242,    Adjusted R-squared:  0.001445 
## F-statistic: 1.132 on 1 and 90 DF,  p-value: 0.2903

## 
## Call:
## lm(formula = alphaNegChoice ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.15491 -0.12216 -0.07414 -0.01518  0.86539 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  0.219194   0.095211   2.302   0.0236 *
## age         -0.005970   0.005118  -1.166   0.2465  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2325 on 90 degrees of freedom
## Multiple R-squared:  0.01489,    Adjusted R-squared:  0.003948 
## F-statistic: 1.361 on 1 and 90 DF,  p-value: 0.2465

## 
## Call:
## lm(formula = alphaPosComp ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.17039 -0.14045 -0.11397 -0.00871  0.82969 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  0.207457   0.100743   2.059   0.0424 *
## age         -0.003459   0.005415  -0.639   0.5246  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.246 on 90 degrees of freedom
## Multiple R-squared:  0.004514,   Adjusted R-squared:  -0.006547 
## F-statistic: 0.4081 on 1 and 90 DF,  p-value: 0.5246

## 
## Call:
## lm(formula = alphaNegComp ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.18426 -0.18099 -0.15342  0.05683  0.80339 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.1787245  0.1175399   1.521    0.132
## age         0.0002398  0.0063179   0.038    0.970
## 
## Residual standard error: 0.287 on 90 degrees of freedom
## Multiple R-squared:  1.6e-05,    Adjusted R-squared:  -0.01109 
## F-statistic: 0.00144 on 1 and 90 DF,  p-value: 0.9698

## 
## Call:
## lm(formula = betaAgency ~ age, data = model_params)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.3948 -3.7399 -0.5663  2.5973 18.5924 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   4.0129     2.2376   1.793   0.0763 .
## age           0.2955     0.1203   2.457   0.0159 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.463 on 90 degrees of freedom
## Multiple R-squared:  0.06287,    Adjusted R-squared:  0.05246 
## F-statistic: 6.038 on 1 and 90 DF,  p-value: 0.01592

## 
## Call:
## lm(formula = betaMachine ~ age, data = model_params)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.8618 -3.3465 -0.7262  2.3170 16.5670 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   4.4853     2.0354   2.204   0.0301 *
## age           0.1644     0.1094   1.502   0.1365  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.97 on 90 degrees of freedom
## Multiple R-squared:  0.02446,    Adjusted R-squared:  0.01362 
## F-statistic: 2.257 on 1 and 90 DF,  p-value: 0.1365

## 
## Call:
## lm(formula = agencyBonus ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.84601 -0.16303 -0.04726  0.04737  1.75107 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.123635   0.171281   0.722    0.472
## age         0.010736   0.009207   1.166    0.247
## 
## Residual standard error: 0.4182 on 90 degrees of freedom
## Multiple R-squared:  0.01488,    Adjusted R-squared:  0.003939 
## F-statistic:  1.36 on 1 and 90 DF,  p-value: 0.2466

Plot relations between model parameters and age

Parameter summary statistics

Mixed-effects beta analysis

## Mixed Model Anova Table (Type 3 tests, S-method)
## 
## Model: estimate ~ ageZ * betaType + (1 | subID)
## Data: betas
##          Effect       df        F p.value
## 1          ageZ 1, 90.00   5.56 *    .021
## 2      betaType 1, 90.00 10.73 **    .001
## 3 ageZ:betaType 1, 90.00     1.16    .284
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: estimate ~ ageZ * betaType + (1 | subID)
##    Data: data
## 
## REML criterion at convergence: 1107.4
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -2.3617 -0.4925 -0.1192  0.3570  3.2407 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  subID    (Intercept) 11.99    3.463   
##  Residual             15.28    3.909   
## Number of obs: 184, groups:  subID, 92
## 
## Fixed effects:
##                Estimate Std. Error      df t value Pr(>|t|)    
## (Intercept)      8.3862     0.4620 90.0000  18.154   <2e-16 ***
## ageZ             1.0920     0.4632 90.0000   2.357   0.0206 *  
## betaType1        0.9439     0.2882 90.0000   3.275   0.0015 ** 
## ageZ:betaType1   0.3115     0.2890 90.0000   1.078   0.2840    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) ageZ  btTyp1
## ageZ        0.000              
## betaType1   0.000  0.000       
## ageZ:btTyp1 0.000  0.000 0.000

Predictor	Estimates	SE	Statistic	p
intercept	8.39	0.46	18.15	<0.001
age	1.09	0.46	2.36	0.019
decision stage	0.94	0.29	3.28	0.001
age x decision stage	0.31	0.29	1.08	0.283

Beta plot

Mixed-effects learning rate analysis

## Mixed Model Anova Table (Type 3 tests, S-method)
## 
## Model: estimate ~ ageZ * valence * agency + (1 | subID)
## Data: learning_rates
##                Effect        df        F p.value
## 1                ageZ  1, 90.00     0.04    .842
## 2             valence 1, 270.00   3.07 +    .081
## 3              agency 1, 270.00     0.25    .617
## 4        ageZ:valence 1, 270.00     0.63    .428
## 5         ageZ:agency 1, 270.00     0.12    .728
## 6      valence:agency 1, 270.00 10.05 **    .002
## 7 ageZ:valence:agency 1, 270.00     2.10    .148
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: estimate ~ ageZ * valence * agency + (1 | subID)
##    Data: data
## 
## REML criterion at convergence: 107.4
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -1.1984 -0.5828 -0.3909  0.0827  3.2210 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  subID    (Intercept) 0.004016 0.06337 
##  Residual             0.065509 0.25595 
## Number of obs: 368, groups:  subID, 92
## 
## Fixed effects:
##                         Estimate Std. Error         df t value Pr(>|t|)    
## (Intercept)             0.170801   0.014889  90.000000  11.472   <2e-16 ***
## ageZ                   -0.002984   0.014909  90.000000  -0.200   0.8418    
## valence1               -0.023387   0.013342 269.999999  -1.753   0.0808 .  
## agency1                 0.006673   0.013342 269.999999   0.500   0.6174    
## ageZ:valence1          -0.010603   0.013360 269.999999  -0.794   0.4281    
## ageZ:agency1            0.004651   0.013360 269.999999   0.348   0.7280    
## valence1:agency1       -0.042297   0.013342 269.999999  -3.170   0.0017 ** 
## ageZ:valence1:agency1  -0.019375   0.013360 269.999999  -1.450   0.1482    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) ageZ  valnc1 agncy1 agZ:v1 agZ:g1 vln1:1
## ageZ        0.000                                          
## valence1    0.000  0.000                                   
## agency1     0.000  0.000 0.000                             
## ageZ:valnc1 0.000  0.000 0.000  0.000                      
## ageZ:agncy1 0.000  0.000 0.000  0.000  0.000               
## vlnc1:gncy1 0.000  0.000 0.000  0.000  0.000  0.000        
## agZ:vlnc1:1 0.000  0.000 0.000  0.000  0.000  0.000  0.000

## 
##  Paired t-test
## 
## data:  model_params$alphaPosChoice and model_params$alphaNegChoice
## t = 3.2464, df = 91, p-value = 0.001636
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  0.05098873 0.21174803
## sample estimates:
## mean difference 
##       0.1313684

## 
##  Paired t-test
## 
## data:  model_params$alphaPosComp and model_params$alphaNegComp
## t = -0.8713, df = 91, p-value = 0.3859
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  -0.12404217  0.04840164
## sample estimates:
## mean difference 
##     -0.03782026

Predictor	Estimates	SE	Statistic	p
intercept	0.17	0.01	11.47	<0.001
age	-0.00	0.01	-0.20	0.842
valence	-0.02	0.01	-1.75	0.080
agency	0.01	0.01	0.50	0.617
age x valence	-0.01	0.01	-0.79	0.428
age x agency	0.00	0.01	0.35	0.728
valence x agency	-0.04	0.01	-3.17	0.002
age x valence x agency	-0.02	0.01	-1.45	0.148

Learning rate plot

Relation between parameter estimates and ‘model-free’ regressions

## 
## Call:
## lm(formula = `(Intercept)` ~ agencyBonus, data = voc_REs_RL)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.9257 -0.6856 -0.0885  0.6727  4.2943 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.03644    0.17338   0.210    0.834
## agencyBonus -0.16590    0.33120  -0.501    0.618
## 
## Residual standard error: 1.324 on 90 degrees of freedom
## Multiple R-squared:  0.00278,    Adjusted R-squared:  -0.0083 
## F-statistic: 0.2509 on 1 and 90 DF,  p-value: 0.6177

## 
## Call:
## lm(formula = zVoC ~ betaAgency, data = voc_REs_RL)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.32488 -0.38100  0.08614  0.47543  1.30577 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.098764   0.133743  -0.738    0.462
## betaAgency   0.007693   0.012301   0.625    0.533
## 
## Residual standard error: 0.6586 on 90 degrees of freedom
## Multiple R-squared:  0.004327,   Adjusted R-squared:  -0.006736 
## F-statistic: 0.3912 on 1 and 90 DF,  p-value: 0.5333

## 
## Call:
## lm(formula = zVoC ~ betaAgency + age, data = voc_REs_RL)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.2932 -0.3993  0.1059  0.4376  1.3427 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -0.55426    0.27046  -2.049   0.0434 *
## betaAgency   0.00164    0.01252   0.131   0.8961  
## age          0.02846    0.01476   1.928   0.0570 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6489 on 89 degrees of freedom
## Multiple R-squared:  0.04426,    Adjusted R-squared:  0.02279 
## F-statistic: 2.061 on 2 and 89 DF,  p-value: 0.1334

## 
## Call:
## lm(formula = zVoC ~ betaAgency + betaMachine, data = voc_REs_RL)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.37387 -0.39512  0.05437  0.43716  1.28471 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.06140    0.14546  -0.422    0.674
## betaAgency   0.01196    0.01391   0.860    0.392
## betaMachine -0.01037    0.01560  -0.665    0.508
## 
## Residual standard error: 0.6607 on 89 degrees of freedom
## Multiple R-squared:  0.009244,   Adjusted R-squared:  -0.01302 
## F-statistic: 0.4152 on 2 and 89 DF,  p-value: 0.6615

Questionnaire relations

DOC

## 
## Call:
## lm(formula = DOC ~ zAge, data = DOC)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -32.234  -6.388  -0.270   7.449  30.317 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   95.527      1.255   76.11   <2e-16 ***
## zAge           2.446      1.274    1.92    0.058 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.97 on 89 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.03978,    Adjusted R-squared:  0.02899 
## F-statistic: 3.687 on 1 and 89 DF,  p-value: 0.05804

## 
## Call:
## lm(formula = DOC ~ zBetaAgency * zAgencyBonus * zAge, data = DOC)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -34.318  -6.665  -0.614   7.416  30.307 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                    95.4327     1.4180  67.300   <2e-16 ***
## zBetaAgency                    -0.6607     1.5865  -0.416   0.6782    
## zAgencyBonus                    1.3423     3.6437   0.368   0.7135    
## zAge                            1.9834     1.4240   1.393   0.1674    
## zBetaAgency:zAgencyBonus        0.9452     2.7639   0.342   0.7332    
## zBetaAgency:zAge                0.7317     1.4672   0.499   0.6193    
## zAgencyBonus:zAge              -6.2496     3.4316  -1.821   0.0722 .  
## zBetaAgency:zAgencyBonus:zAge  -3.1118     2.5602  -1.215   0.2276    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.95 on 83 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.1082, Adjusted R-squared:  0.03295 
## F-statistic: 1.438 on 7 and 83 DF,  p-value: 0.2013

LOC

## 
## Call:
## lm(formula = LOC ~ zAge, data = LOC)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.4335 -3.3923 -0.4242  3.4805 10.1914 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  12.6288     0.4372  28.886   <2e-16 ***
## zAge          0.2453     0.4392   0.559    0.578    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.17 on 89 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.003494,   Adjusted R-squared:  -0.007703 
## F-statistic: 0.3121 on 1 and 89 DF,  p-value: 0.5778

## 
## Call:
## lm(formula = LOC ~ zBetaAgency * zAgencyBonus * zAge, data = LOC)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.9583 -3.4626 -0.3911  3.2787 10.0915 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                   12.92475    0.50467  25.610   <2e-16 ***
## zBetaAgency                   -0.14273    0.56284  -0.254    0.800    
## zAgencyBonus                   1.41264    1.29225   1.093    0.277    
## zAge                           0.36724    0.50171   0.732    0.466    
## zBetaAgency:zAgencyBonus       1.23889    0.98011   1.264    0.210    
## zBetaAgency:zAge              -0.02981    0.52015  -0.057    0.954    
## zAgencyBonus:zAge             -1.01683    1.21653  -0.836    0.406    
## zBetaAgency:zAgencyBonus:zAge -0.31812    0.90795  -0.350    0.727    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.238 on 83 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.04037,    Adjusted R-squared:  -0.04056 
## F-statistic: 0.4988 on 7 and 83 DF,  p-value: 0.8329

BDI

## 
## Call:
## lm(formula = zBDI ~ zAge, data = BDI)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.94728 -0.78671 -0.01517  0.72806  2.78555 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.122e-16  1.042e-01   0.000    1.000
## zAge        3.587e-02  1.048e-01   0.342    0.733
## 
## Residual standard error: 0.9993 on 90 degrees of freedom
## Multiple R-squared:  0.001301,   Adjusted R-squared:  -0.009796 
## F-statistic: 0.1172 on 1 and 90 DF,  p-value: 0.7329

## 
## Call:
## lm(formula = zBDI ~ zBetaAgency * zAgencyBonus * zAge, data = BDI)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.85715 -0.65938  0.00033  0.69522  2.67871 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)
## (Intercept)                    0.08489    0.11978   0.709    0.480
## zBetaAgency                    0.20766    0.13439   1.545    0.126
## zAgencyBonus                   0.34329    0.30858   1.113    0.269
## zAge                           0.04168    0.11917   0.350    0.727
## zBetaAgency:zAgencyBonus       0.28280    0.23403   1.208    0.230
## zBetaAgency:zAge              -0.02180    0.12418  -0.176    0.861
## zAgencyBonus:zAge              0.07767    0.29050   0.267    0.790
## zBetaAgency:zAgencyBonus:zAge  0.11565    0.21679   0.533    0.595
## 
## Residual standard error: 1.012 on 84 degrees of freedom
## Multiple R-squared:  0.04419,    Adjusted R-squared:  -0.03546 
## F-statistic: 0.5549 on 7 and 84 DF,  p-value: 0.7903

STAI

## 
## Call:
## lm(formula = zSTAI_t ~ zAge, data = STAI)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.82245 -0.96538  0.01261  0.83118  2.16747 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.001085   0.104658   0.010    0.992
## zAge        0.060134   0.106243   0.566    0.573
## 
## Residual standard error: 0.9982 on 89 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.003587,   Adjusted R-squared:  -0.007609 
## F-statistic: 0.3204 on 1 and 89 DF,  p-value: 0.5728

## 
## Call:
## lm(formula = zSTAI_t ~ zBetaAgency * zAgencyBonus * zAge, data = STAI)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.84138 -0.92206 -0.05708  0.79479  2.23553 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)
## (Intercept)                    0.03420    0.12090   0.283    0.778
## zBetaAgency                    0.15007    0.13569   1.106    0.272
## zAgencyBonus                   0.18969    0.31114   0.610    0.544
## zAge                           0.02867    0.12130   0.236    0.814
## zBetaAgency:zAgencyBonus       0.16540    0.23606   0.701    0.485
## zBetaAgency:zAge              -0.08046    0.12645  -0.636    0.526
## zAgencyBonus:zAge             -0.05753    0.29196  -0.197    0.844
## zBetaAgency:zAgencyBonus:zAge -0.09840    0.21793  -0.452    0.653
## 
## Residual standard error: 1.016 on 83 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.03663,    Adjusted R-squared:  -0.04462 
## F-statistic: 0.4508 on 7 and 83 DF,  p-value: 0.867

## 
## Call:
## lm(formula = zSTAI_s ~ zAge, data = STAI)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9203 -0.6732 -0.1498  0.4769  3.1426 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.937e-16  1.033e-01   0.000    1.000
## zAge        1.368e-01  1.038e-01   1.318    0.191
## 
## Residual standard error: 0.9905 on 90 degrees of freedom
## Multiple R-squared:  0.01894,    Adjusted R-squared:  0.008035 
## F-statistic: 1.737 on 1 and 90 DF,  p-value: 0.1909

## 
## Call:
## lm(formula = zSTAI_s ~ zBetaAgency * zAgencyBonus * zAge, data = STAI)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9560 -0.6394 -0.1377  0.5929  3.2736 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)
## (Intercept)                    0.10180    0.11729   0.868    0.388
## zBetaAgency                    0.18433    0.13159   1.401    0.165
## zAgencyBonus                   0.34019    0.30216   1.126    0.263
## zAge                           0.14574    0.11669   1.249    0.215
## zBetaAgency:zAgencyBonus       0.36647    0.22917   1.599    0.114
## zBetaAgency:zAge              -0.09988    0.12160  -0.821    0.414
## zAgencyBonus:zAge             -0.07798    0.28446  -0.274    0.785
## zBetaAgency:zAgencyBonus:zAge -0.02332    0.21228  -0.110    0.913
## 
## Residual standard error: 0.9909 on 84 degrees of freedom
## Multiple R-squared:  0.08352,    Adjusted R-squared:  0.007147 
## F-statistic: 1.094 on 7 and 84 DF,  p-value: 0.3748